Summary
That the central theorem is valid for (forward) martingales is a result with a long history, beginning with Lévy [6], the most refined and recent results being due to Billingsley [3].
Not altogether surprisingly, an analogous result holds for backwards martingales, and the proof, which parallels closely that of Billingsley, occupies Section 1. Examples are given in Section 2.
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Loynes, R.M. The central limit theorem for backwards martingales. Z. Wahrscheinlichkeitstheorie verw Gebiete 13, 1–8 (1969). https://doi.org/10.1007/BF00535793
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DOI: https://doi.org/10.1007/BF00535793